Loehr–Warrington rational $q$-Catalan partition formula conjecture

Fix $a,b\in\mathbb{N}$ with $\gcd(a,b)=1$. Let $\mathrm{D}(a,b)$ be the set of partitions whose diagrams fit in the triangle with vertices $(0,0)$, $(0,a)$, and $(b,a)$. For a cell $c$, let $\mathsf{arm}(c)$ and $\mathsf{leg}(c)$ denote its arm and leg, and let $h^+_{b,a}(\mu)$ count cells satisfying $-a<a\mathsf{arm}(c)-b\mathsf{leg}(c)\leq b$. Rational $q$-Catalan formula conjecture.

$$\mathsf{Cat}_{a,b}(q)=\sum_{\mu\in\mathrm{D}(a,b)}q^{|\mu|+h^+_{b,a}(\mu)}.$$

This proposes a partition-statistic interpretation of the rational $q$-Catalan number; the supplied text does not give a proof or resolution.